Spectroscopic Instruments 95 Spectroscopic Instruments by division of amplitude Mach-Zehnder (division of amplitude) Michelson Fringe localisation LIGO Fabry-Perot (FPI) Multi-layer coatings 96
Mach-Zehnder Interferometer Interference fringes as a function of δ u 0 2 (e i 2 e i 2 ) = u 0 i sin( /2) 1 p 2 u 0 1 p 2 u 0 e i(kl+ ) r = t =1/ p 2 u 0 2 (e i 2 + e i 2 ) = u 0 cos( /2) 1 p 2 u 0 e i(kl+ ) u 0 1 p 2 u 0 note: π phase jump upon reflection of the denser medium 97 Michelson Interferometer Fourier-transform spectroscopy Michelson-Morley experiment Gravitational wave detection LIGO Hanford 98
Michelson Interferometer beam splitter path difference x =2d compensation plate d 99 Fourier-Transform Spectroscopy a ae ikx I out (x) = u u = a + ae ikx 2 = I 0 (1 + cos kx)/2 d = x/2 100-1
Fourier-Transform Spectroscopy F (x) =I(x) I 0 /2 / cos kx a ae ikx 1 1 2 3 4 5 x =2d d = x/2-1 100-3 Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx S( ) F (x) / cos(k kx 0 x) cos( kx/2) x 0 / k 101-1
Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx S( ) F (x) / cos(k 0 x) cos( kx/2) x 0 / k 101-2 Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx 101-3
Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx 101-4 Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx 101-5
Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx 101-6 Fourier-Transform Spectroscopy I 1 (x) scanning Michelson I 2 (x) x I 1 (x) =I 0 /2(1 + cos k 1 x) x I 2 (x) =I 0 /2(1 + cos k 2 x) x max I(x) I(x) = h u 1 (x)+u 2 (x) 2 i t = u 1 (x) 2 + u 2 (x) 2 /(k 2 k 1 ) x +2hRe(u 1(x)u 2 (x)i t = I 1 (x)+i 2 (x) 102
Instrument Width & Resolving Power The envelope varies slowly in x for at least one period over the full range, we need Instrumental Width cos 1 2 (k 1 k 2 )x = cos x x max Inst = min =1/x max Resolving Power mean Inst = = x max 103 Finite Coherence Length wave packets of finite duration fringes no interference 104
Finite Coherence Length F (x) =I(x) I 0 /2 /ha (x 0 )a(x + x 0 )i x 0 cos kx a ae ikx 1 d = x/2 1 2 3 4 5-1 wave trains of finite length l c = c t c 105-1 Finite Coherence Length F (x) =I(x) I 0 /2 /ha (x 0 )a(x + x 0 )i x 0 cos kx a ae ikx 1 1 2 3 4 5 x =2d d = x/2-1 wave trains of finite length l c = c t c 105-2
Autocorrelation Measurements wave packets of finite coherence length F (x) /ha (x 0 )a(x + x 0 )i x 0 l c x l c x visibility I min (x) = I max = ha (x 0 )a(x + x 0 )i x0 I max + I min ha (x 0 )a(x 0 )i x 0 106 Michelson-Morley Experiment Light-carrying ether? Speed of light invariant? Michelson-Morley experiment! universal & constant speed of light pattern unchanged! 107-3
Fringe Localisation 108 Reflection of Parallel Plates = kx sin = kx cos + = /2 Rotationally symmetric around P-P axis Young s fringe to axis Etalon fringe (Michelson interferometer) to axis 109
Parallel Plates Michelson Interferometer d I / cos 2 ( 1 2kx cos ) x =2d equivalent to Michelson! 110-2 Fringe Localisation point-like (image) sources double slit fringes fringes are not localised extended light sources fringes localised at infinity 111
Fringe Localisation point source fringes not localised 112 Fringe Localisation point source fringes not localised 113
Fringe Localisation extended source 114-1 extended source Fringe Localisation fringes localised at infinity 114-2
Fringe Localisation Parallel reflecting surfaces extended source images t source Extended source path difference xcos 2t=x Fringes localized at infinity Circular fringe constant 2t=x circular fringe constant Fringes of equal inclination 115-1 Parallel reflecting surfaces extended source Fringe Localisation images t source fringes localised at infinity Extended source path difference xcos 2t=x Fringes localized at infinity Circular fringe constant 2t=x circular fringe constant Fringes of equal inclination 115-2
Newton s Rings from Wikipedia 116 soap bubbles 117
fringes everywhere Fringe Localisation Double Slit Plane waves (Fresnel diffraction) double-slit fringes at infinity (Fraunhofer) 118 Fringe Localisation Grating fringes at infinity Plane waves Z Fraunhofer grating 119
Fringe type and Localisation Wedged Parallel Point Source Non-localised Equal thickness Non-localised Equal inclination Extended Source Localised in plane of Wedge Equal thickness Localised at infinity Equal inclination 120 LIGO Laser Interferometric Gravitational Wave Observatory 4 Km 121
LIGO Laser Interferometric Gravitational Wave Observatory precision L/L 10 21 122 Fabry-Perot Interferometer FPI or etalon 123
Fabry-Perot Interferometer multiple beam interference U 0 t 1 t 2 (r 1 r 2 ) 3 e 3i U 0 t 1 t 2 (r 1 r 2 ) 2 e 2i U 0 t 1 r 1 r 2 U0 t 1 t 2 r 1 r 2 e i =2kd cos 124 Fabry-Perot Interferometer Michelson FPI 125
Fabry-Perot Interferometer FPI transmission 126 = k2d cos Transmission Fabry-Perot Interferometer F = R 1 R Finesse F = peak separation FWHM of one peak = k2d cos 127
Scanning Fabry-Perot Piezo or pressure tuning scan the optical path-length difference 128 FPI as Spectrometer 129
FPI as Spectrometer interlacing rings from the Oxford Physics Webpage 130 FPI as Spectrometer 131
FPI Finesse Spectroscopic tool: 10... 100 Laser cavities: 100... 1000 Trapping light: 10,000... 1,000,000 (for studying quantum effects at the single-photon level) Chromatic Resolving Power Fabry-Perot: F = FSR FWHM = (2d) 1 p =2d p F = = Grating: p = d sin p dispersion: minimum: p = d p cos p (Np+ 1) = Nksin( p + )d/2 cos p = /(Nd) p N = = =